Topological Characterization of Quantum Phase Transitions in a Spin-$1/2$ Model

We have introduced a novel Majorana representation of $S=1/2$ spins using the Jordan-Wigner transformation and have shown that a generalized spin model of Kitaev defined on a brick-wall lattice is equivalent to a model of noninteracting Majorana fermions with $Z_2$ gauge fields without redundant degrees of freedom. The quantum phase transitions of the system at zero temperature are found to be of topological type and can be characterized by nonlocal string order parameters (SOP). In appropriate dual representations, these SOP become local order parameters and the basic concept of Landau theory of continuous phase transition can be applied.

Figure 1

A brick-wall lattice (c) with its equivalent honeycomb lattice (d). (a) and (b) are the one- and two-row limits of the brick-wall lattices, respectively.

Figure 2

The second derivative of the ground state energy density $E_0$ (a) and the nonlocal SOP (b) for a single chain. The arrows in (b) indicate the RG flow.

Figure 3

(a) The phase diagram with RG flows for the two-leg ladder. (b) The second derivative of $E_0$ along the dotted line in (a). (c) and (d) are two deformed single-chain representations of the two-leg ladder.

Figure 4

(a) The six-row brick-wall lattice and (b) its equivalent three-row lattice.

Figure 5

The phase diagram with quantum critical lines of the $2M$-row brick-wall lattice. In the $M=\infty$ panel, the four phases are, respectively, labeled by ($A_x$, $A_y$, $A_z$) and $B$ following the notation of Kitaev [5].